Wild Maths encourages students to explore maths beyond the classroom and is designed to nurture mathematical creativity. The site is aimed at 7 to 16 year-olds, but open to all. It provides games, investigations, stories and spaces to explore, where discoveries are to be made. Some have starting points, some a big question and others offer you a free space to investigate.

Can you fold a piece of paper in half? Of course you can, it's easy, you just match the two corners along one side. But can you fold it in thirds? You might be able to with a bit of fiddling and guessing, but what about fifths? Or sevenths? Or thirteenths? There is a simple way you can fold a piece of paper into any fraction you would like – exactly – no guessing or fiddling needed!

This article was inspired by content on Wild Maths, which encourages students to explore maths beyond the classroom and is designed to nurture mathematical creativity. The site is aimed at 7 to 16 year-olds, but open to all. It provides games, investigations, stories and spaces to explore, where discoveries are to be made. Some have starting points, some a big question and others offer you a free space to investigate.

Sometimes real progress in maths comes when you find a way of looking at a problem in two different ways. Here is a great example of this.

Suppose you have people in a room and each person shakes hands with each other person once. How many handshakes do you get in total? The first person shakes hands with other people, the second shakes hands with the remaining people, the third shakes hands with remaining people, etc, giving a total of

handshakes.

But we can also look at this in another way: each person shakes hands with others and there are people, giving handshakes. But this counts every handshake twice, so we need to divide by 2, giving a total of

handshakes.

Putting these two arguments together, we have just come up with the formula for summing the first integers and we’ve proved that it is correct:

This puzzle is inspired by content on our sister site Wild Maths, which encourages students to explore maths beyond the classroom and is designed to nurture mathematical creativity. The site is aimed at 7 to 16 year-olds, but open to all. It provides games, investigations, stories and spaces to explore, where discoveries are to be made. Some have starting points, some a big question and others offer you a free space to investigate.

Only three things in life are certain: death, taxes and parking fees. But even a menacing parking meter is an
excuse to do some maths.

Imagine, for example, that the car park costs £1.50. The machine only accepts 10p and 20p coins.
There are obviously different ways of putting the money into the car park machine, for example

10p, 10p, 20p, 20p, 10p, 10p, 10p, 10p, 20p, 10p, 10p, 10p

or

10p, 10p, 10p, 10p, 20p, 20p, 20p, 10p, 20p, 20p.

You could probably go for the rest of the month without feeding the machine in the same way twice.
Can you feed the machine in a different way each day of the year?

You can find a longer version of this puzzle, including some follow-up questions to investigate, on the Wild Maths site. Wild Maths encourages students to explore maths beyond the classroom and is designed to nurture mathematical creativity. The site is aimed at 7 to 16 year-olds, but open to all. It provides games, investigations, stories and spaces to explore, where discoveries are to be made. Some have starting points, some a big question and others offer you a free space to investigate.

One interpretation of the strange theory of quantum mechanics is that
tiny particles can simultaneously exist in states that we would
usually deem mutually
exclusive. For example, an electron can be in two places at once, or a
radioactive atom can be both decayed an non-decayed at the same
time. It's only when we go to measure a system in
superposition, as this strange state is called, that reality
somehow "collapses" to one of the possibilities.

In 1935 the physicist Erwin Schrödinger, who made major
contributions to the theory of quantum mechanics, developed a thought experiment in order to demonstrate just how
counter-intuitive the idea of superposition is. We let him describe it
in his own words, taken from a translation of his 1935 paper:

One can even set up quite ridiculous cases. A cat is penned up
in a steel chamber, along with the following device (which must be
secured against direct interference by the cat): in a Geiger counter
there is a tiny bit of radioactive substance, so small, that perhaps
in the course of the hour one of the atoms decays, but also, with
equal probability, perhaps none; if it happens, the counter tube
discharges and through a relay releases a hammer which shatters a
small flask of hydrocyanic acid.

Thus, when an atom decays, poison will be released from the flask and the cat
killed. And here's the main point. If it is true that, as long as we
don't look, the system can evolve into a superposition state of atoms being simultaneously decayed and not decayed, then
it follows that, as long as we don't look, the cat will be simultaneously
dead and alive. Poor cat. Or should we say lucky cat?

John D. Barrow, mathematician, cosmologist and boss of Plus, explores maths and the arts in a public talk in Cambridge on Monday, 02 November 2015.
Barrow will look at ways in which maths can shed light upon a range of questions in the arts and how problems of art and design inspire new mathematical questions. The canvas will be broadly drawn with examples from different areas of the arts, including painting, textual analysis, diamond cutting, Henry Moore's stringed figures, ballet, and even the best place to stand when viewing statues.